Gauge R&R against Cpk

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Two sliders. Watch measurement error widen the curve you actually see, take a bite out of your Cpk, and start rejecting good parts.

MEASURED VARIATION = ACTUAL + MEASUREMENT SYSTEM · FREE · NOTHING YOU TYPE LEAVES YOUR BROWSER

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What gauge R&R does to Cpk

MEASURED VARIATION = ACTUAL + MEASUREMENT SYSTEM

Variances add. The spread you measure is the process spread and the gauge spread together, and Cpk is calculated from what you measured — so every Cpk you have ever reported was lower than the process deserved.

TRUE Cpk 1.33 0.502.50

GAUGE R&R 30% 0%30%80%

Actual process variation Measured variation What the gauge added Specification limits

Wrong decisions, per million parts

Why the curve moves so little at first. Variances add, not standard deviations, so the gauge has to be large before it shifts the total much. A 30 percent gauge widens the measured curve by under five percent. Push the slider past fifty and the two curves pull apart in a hurry — that is quadrature, arriving late and all at once.

The arithmetic. With %GRR expressed against total variation (the slider above; the table below uses %GRR against tolerance, which is why its numbers differ), observed Cpk works out to exactly √(1 − %GRR²) times the true Cpk. Ten percent costs you half a percent of Cpk, thirty costs you five, fifty costs thirteen, and eighty costs you forty.

For educational use. This page is a study aid. The method behind it has far more depth and detail than one page and one tool can cover, and what you see here is not everything the method offers or everything this site offers. For full understanding and application, use the governing standards and texts, and seek further education, training or qualified advice. Results depend on the data you enter, so check them before relying on them.

You never see the process. You see the process plus the gauge.

Every measurement carries the part's real variation and the measurement system's variation together. They add as variances, so what you observe is wider than what exists:

σ²_observed = σ²_process + σ²_measurement

Which means the Cpk you calculate is not your process capability. It is your process capability after the gauge has had its share. Improve the gauge and the Cpk rises without a single thing changing on the machine.

The same gauge error costs you two completely different things

This is the part that is not obvious, and it falls straight out of the arithmetic:

COMPUTED — 0.040 TOLERANCE, CENTERED PROCESS, 1,000,000 SIMULATED PARTS
True Cpk	%GRR (of tolerance)	Cpk you observe	Good parts rejected	Bad parts accepted
1.00	10	1.00	0.04%	11.0%
1.00	30	0.96	0.20%	23.9%
1.00	50	0.89	0.55%	31.4%
1.33	30	1.24	0.02%	31%
1.67	10	1.65	0.00%	~0 (under 1 ppm)
1.67	30	1.49	0.00%	~0 (under 1 ppm)
1.67	50	1.28	0.01%	~0 (under 1 ppm)

Read down the Cpk 1.67 rows: a 30 percent gauge costs you 0.18 of Cpk — enough to fail a customer requirement — and lets essentially nothing through, because a capable process barely puts parts near the limits.

Now read the Cpk 1.00 rows: the same 30 percent gauge barely moves the Cpk at all, from 1.00 to 0.96, and quietly passes nearly a quarter of the bad parts.

On a capable process, gauge error shows up in your Cpk. On a marginal one, it shows up in your customer's inbox. If you monitor only the capability number, the gauge problem is invisible in exactly the case where it is costing you the most.

Why the acceptance criteria are what they are

The conventional bands — under 10 percent acceptable, 10 to 30 percent conditional, over 30 percent not acceptable — look like round numbers chosen by committee. The table shows why they are not unreasonable. At 10 percent the effects above are small in both directions. By 30 percent you are losing real capability and passing real defects. Past that, you are measuring the gauge.

What to do with a gauge you cannot improve

Guard band. Tighten the acceptance limits inside the specification by some fraction of the measurement error. You will reject more good parts on purpose in order to stop passing bad ones — an explicit trade rather than an accidental one.
Average repeated readings. Measuring three times and averaging cuts the repeatability standard deviation by a factor of √3 (reproducibility between appraisers is not reduced). Slow, but it costs nothing in capital.
Fix the process instead. Lifting true Cpk from 1.00 to 1.67 makes the escape problem disappear on its own, as the table shows. Sometimes that is the cheaper project.
Stop arguing about a number you cannot measure. A supplier reporting Cpk from a 40 percent gauge is reporting a fiction — almost certainly an understatement of the process, and an unknown escape rate. Settle the measurement system first.

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